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arXiv:quant-ph/0606056

Application of a renormalization algorithm in Hilbert space to the study of many-body quantum systems

Tarek Khalil, J. Richert

Published Jun 7, 2006• 1 citation
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Abstract

We implement an algorithm which is aimed to reduce the number of basis states spanning the Hilbert space of quantum many-body systems. We test the efficiency of the procedure by working out and analyzing the spectral properties of strongly correlated and frustrated quantum spin systems. The role and importance of symmetries are investigated

DOI
10.48550/arxiv.quant-ph/0606056

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We implement an algorithm which is aimed to reduce the number of basis states spanning the Hilbert space of quantum many-body systems. We test the efficiency of the procedure by working out and analyzing the spectral properties of strongly correlated and frustrated quantum spin systems. The role and importance of symmetries are investigated
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size sis_{i}si​ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
A biorthonormal-block density-matrix renormalization group algorithm is proposed to accurately compute properties of large-scale non-Hermitian many-body systems, in which a renormalized-space partition of the non-Hermitian reduced density matrix is implemented to fulfill the prerequisite for the biorthonormality of the renormalization group (RG) transformation and to optimize the construction of saved Hilbert spaces. A redundancy in saved spaces of the reduced density matrix is exploited to reduce a condition number resulting from the nonunitarity of the left and right transformation matrices, in order to ensure the numerical stability of the RG procedure. The algorithm is successfully applied to an interacting fermionic Su-Schrieffer-Heeger model with nonreciprocal hoppings and staggered complex chemical potential, exhibiting novel many-body phenomena.
The exploration of neural network quantum states has become widespread in studies of complicated quantum many-body systems. However, achieving high precision remains challenging due to the exponential growth of Hilbert space size and the intricate sign structures. Utilizing symmetries of the physical system, we propose a method to evaluate and sample the variational Ansatz within a symmetric subspace. This approach isolates different symmetry sectors, reducing the relevant Hilbert space size by a factor approximately proportional to the size of the symmetry group. It is inspired by exact diagonalization techniques and the work of Choo et al., [Phys. Rev. Lett. 121, 167204 (2018)]. We validate our method using the frustrated spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg chain, and we compare its performance to the case without symmetrization. The results indicate that our symmetric subspace approach achieves a substantial improvement over the full Hilbert space on optimizing the Ansatz, reducing the energy error by orders of magnitude. We also compare the results on degenerate eigenstates with different quantum numbers, highlighting the advantage of operating within a smaller Hilbert subspace.
On propose un algorithme non perturbatif pour étudier les états de basse énergie des spectres de systèmes quantiques fortement corrélés à température nulle dans un espace de Hilbert réduit. On utilise une technique de projection qui permet de générer des Hamiltoniens effectifs dans des espaces de Hilbert réduits en renormalisant les constantes de couplage qui caractérisent l'Hamiltonien. On teste la procédure en calculant et analysant les propriétés spectrales des systèmes antiferromagnétiques, échelles de spins à deux montants dans lesquels une approche de perturbation n'est pas valide. Le rôle et l'importance des symétries sont étudiés. On développe une approche qui identifie les points exceptionnels dans le spectre d'énergie des systèmes qui présentent des transitions de phase quantiques. Ces transitions sont liées aux points fixes dans l'espace de Hilbert dans le cadre de la théorie de renormalisation. On examine cette approche dans le cadre de l'étude des transitions du premier ordre et des croisements évités.